# Happy new year 2019 digit problem

#### apple2357

##### Junior Member
The number 2019, its double 4038, and its triple 6057 contain all 10 digits. I am interested in the next number with this property?
I have found one but am not sure if there is one before it. My approach has been trial and error but feel there ought to be some kind of strategy. I am guessing the second digit 0?

Any thoughts?

#### Jomo

##### Elite Member
The number 2019, its double 4038, and its triple 6057 contain all 10 digits. I am interested in the next number with this property?
I have found one but am not sure if there is one before it. My approach has been trial and error but feel there ought to be some kind of strategy. I am guessing the second digit 0?

Any thoughts?
Which number did you find? How do you suppose to show that there is not one before it?

#### Denis

##### Senior Member

1692 3384 5076
1809 3618 5427
1902 3804 5706
1908 3816 5724
1920 3840 5760
2019 4038 6057
2079 4158 6237
2169 4338 6507
2190 4380 6570
2673 5346 8019
2697 5394 8091
2703 5406 8109
2730 5460 8190
2967 5934 8901
3027 6054 9081
3078 6156 9234
3267 6534 9801
3270 6540 9810

Brute strength. I see no way to nicely "calculate".

#### Jomo

##### Elite Member

1692 3384 5076
1809 3618 5427
1902 3804 5706
1908 3816 5724
1920 3840 5760
2019 4038 6057
2079 4158 6237
2169 4338 6507
2190 4380 6570
2673 5346 8019
2697 5394 8091
2703 5406 8109
2730 5460 8190
2967 5934 8901
3027 6054 9081
3078 6156 9234
3267 6534 9801
3270 6540 9810

Brute strength. I see no way to nicely "calculate".
The sum of the digits of all the numbers in column 3 add to 18. Why is that?

#### Denis

##### Senior Member
The sum of the digits of all the numbers in column 3 add to 18. Why is that?
No idea...good stuff for your "whiteboard"

Seems to be for that "case" only; tried a few others,
like 2789: 2789*3 = 8367: sum digits = 24

#### Jomo

##### Elite Member
No idea...good stuff for your "whiteboard"

Seems to be for that "case" only; tried a few others,
like 2789: 2789*3 = 8367: sum digits = 24
But this is not the special number that fits our case.

#### Denis

##### Senior Member
But this is not the special number that fits our case.
That's what I meant with "Seems to be for that "case" only".

I was unclearly saying that the "18" result is only when all 10 digits appear.

#### apple2357

##### Junior Member
Which number did you find? How do you suppose to show that there is not one before it?

The one i got was 3027! But it was mostly guess work!

#### Denis

##### Senior Member
Any "reason" you posted this?

#### apple2357

##### Junior Member
Any "reason" you posted this?
Someone sent it to me. And i couldn't see an interesting way to approach it so wondered if i was missing something!

#### Jomo

##### Elite Member
That's what I meant with "Seems to be for that "case" only".

I was unclearly saying that the "18" result is only when all 10 digits appear.
Denis, can you please do me a favor and run your program for more numbers of this type?

#### Subhotosh Khan

##### Super Moderator
Staff member
Denis, can you please do me a favor and run your program for more numbers of this type?
If you play LOTTO with those numbers - he can demand a cut of the winnings!!

#### Jomo

##### Elite Member
If you play LOTTO with those numbers - he can demand a cut of the winnings!!
He really is a demanding person, isn't he?

Happy New Years!!

#### Denis

##### Senior Member
Denis, can you please do me a favor and run your program for more numbers of this type?
There's no more.

Didn't write a program: calculated those in my head during my morning walk :lol:

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#### Jomo

##### Elite Member
1234567890 2469135780 3703703670-----sum of digits of the last number is 36, which is 2*18. Strange

9012345678 ..... ..... sum of digits of the last number is 36, which is 2*18. Strange

If this pattern of multiples of 18 will continue, then I will like to try to prove this with minimal help. So if someone sees the prove, can you please give me some leading hints.

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#### Jomo

##### Elite Member
There's no more.

Didn't write a program: calculated those in my head during my morning walk :lol:

#### Denis

##### Senior Member
What do you mean Steve?
There are no more cases of 3 4digit numbers
containing all the 10 digits (at least once each),
where (1st number = a) the 3 numbers are a, 2a and 3a.

#### Jomo

##### Elite Member
What do you mean Steve?
There are no more cases of 3 4digit numbers
containing all the 10 digits (at least once each),
where (1st number = a) the 3 numbers are a, 2a and 3a.
I guess what you are saying is true but there are 5, 6, 7, .... digit numbers which you did not find. This really seems like an interesting problem even if does not go on for ever and I'd like to look into this. After all, this is what real mathematicians do (not that I am one).

#### Denis

##### Senior Member
I guess what you are saying is true but there are 5, 6, 7, .... digit numbers which you did not find.
BUT I was NOT looking for those: I was looking for 3 numbers that are
each 4 digits, NOTHING else. Which meant the lower was in range
1000 to 3333, meaning high case of 3333,6666,9999.
Sorry 'bout dat, chief...as Maxwell Smart would say...

#### Jomo

##### Elite Member
BUT I was NOT looking for those: I was looking for 3 numbers that are
each 4 digits, NOTHING else. Which meant the lower was in range
1000 to 3333, meaning high case of 3333,6666,9999.
Sorry 'bout dat, chief...as Maxwell Smart would say...
Look Jackson, I am asking you for a favor. Can you modify your morning walk a bit, please?